Stochastic orders and their applications in financial optimization

Masaaki Kijima * Masamitsu Ohnishi t

Tokyo Metropolitan University Osaka University

April 12, 1999

Abstract

Stochastic orders and inequalities are very useful tools in various areas of economics

and finance. The purpose of this paper is to describe main results obtained so far by

using the idea of stochastic orders in financial optimization. Especially, the emphasis

is placed on the demand and shift effect problems in portfolio selection. Some other

examples, which are not related directly to optimization problems, are also given to

demonstrate the wide spectrum of application areas of stochastic orders in finance.

Keywords: portfolio selection, demand problem, shift effect problem, bivariate

characterization, risk aversion, generalized harmonic mean, equilibrium price, Markov

chain.

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1 Introduction

Stochastic orders and inequalities have been used in many diverse areas of probability and

statistics. For example, consider a single-server queueing system with IFR (increasing failure

rate) service times and a work-conserving service discipline. Let ‘7, denote the ergodic

sojourn time of a customer when service discipline ?r is used. Then, Hirayama and Kijima

(1989) proved that T&o is stochastically smaller than the other T, in the increasing convex

order. That is, the ergodic sojourn time in G/IFR/l queue is minimized by FIFO (first in,

first out) discipline in the sense that

for any non-decreasing and convex function f. Hence, if the cost function f is non-decreasing

and convex, then the expected cost incurred from customers’ sojourn times is minimized by

FIFO discipline. Such a result has an apparent importance in practice from managerial

points of view. Many other examples that use stochastic orders as a key tool are found in

Shaked and Shanthikumar (1994), where some commonly used stochastic orders in applied

probability are also presented.

Tuning to finance areas, future prices (or rates of return) of financial assets are described

by random variables. Hence, financial optimization problems cannot be an exception for the

areas to which stochastic orders are applicable. For example, stochastic orders have been

proved a very useful tool in portfolio selection. Consider an investor with von Neumann-

Morgenstern utility function u who wants to invest his/her wealth into one of two assets, A

or B say. Let X and Y be random variables representing the outcomes of assets A and B,

respectively. Then, according to the expected utility principle, the investor prefers asset A

to B if and only if

But, if the inequality (1.2) holds for all investors whose utility functions belong to some

function class, then this is just the notion of stochastic orders for X and Y as described in

(1.1). The purpose of this paper is to present stochastic orders useful in financial optimization

problems and to illustrate their applicability. However, we do not intend to describe the

state of the art of this area. Rather, the emphasis is placed on the demand and shift effect

problems in portfolio selection, in order to show some techniques, which have been developed

in applied probability and operations research for stochastic optimization problems. Some

other examples, which are not related directly to optimization problems, are also given to

demonstrate the wide spectrum of application areas of stochastic orders in finance.

This paper is organized as follows. In the next section, we describe the portfolio selection

problem formally, and define the notion of generalized harmonic mean for subsequent devel-

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opments in this paper. Section 3 presents stochastic orders useful in finance and provides

a concise review of their important properties. Section 4 considers the demand problem

while, in Section 5, the shift effect problem is discussed in some details. These problems are

shown to have a bivariate character of their own, and hence the bivariate characterization of

stochastic orders play an important role in these sections. Finally, in Section 6, we present

two other examples which hopefully demonstrate that the stochastic orders are a promising

toot in various areas of finance.

2 The portfolio selection problem

Consider an investor with initial investment capital W = 1. Suppose that there are n invest-

ment opportunities Xr , Xs, . , X,, which are represented as (not necessarily nonnegative)

random variables. The set of random variables under consideration is called the choice set.

We study a single period portfolio problem of allocating the wealth to the n investments to

maximize the expected utility of the resulting final wealth. The portfolio selection problem

can be written as

~$E[u($a~xi)]~ (2.1)

where u denotes a von Neumann-Morgenstern utility function on lR = (--00,~) of the

investor, the expectation is taken over the n random variables, and A denotes some constraint

in R”. Typically, if no short sales are allowed then the constraint is given by

A= a=(q,q,... i

,o”):~o,=l, ai>0 , 1

(2.2) 1=1

or A = R” if no constraint is imposed. Throughout this paper, the utility function u

is assumed to satisfy u’ > 0 and u” 5 ‘0, i.e., the investor is risk averse, unless stated

otherwise. This is so, because investors prefer more to less, and a certain outcome is usually

preferred to uncertain ones, provided the mean is the same. Also, to avoid unnecessary

technical difficulties, we assume that the expectations under consideration always exist.

A portfolio is a random variable of the form

P = eaiX;j (al,a2,... , Q,) E d. i=I

An optimal portfolio is a solution of the portfolio problem (2.1). We will refer to the optimal

fraction to invest in the ith asset as a:. The portfolio problem (2.1) has been extensively

studied under a variety of settings in the literature. As examples, see Hadar and Seo (1988,

1990), Jewitt (1987, 1989), Kijima and Ohnishi (1993), M ar k owitz (1959), McEntire (1984),

Meyer and Ormiston (1985,1994) and Wright (1987). S ee also Eeckhoudt and Gollier (1995),

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Gollier (1995, 1997) and Sakagami (1997) an d f re erences therein for recent developments in

this area.

When X, are normally distributed, the portfolio selection problem is reduced to the

Markowitz model (1959). In this case, the portfolio P is also normally distributed and,

denoting the covariance by

Uij=COV[Xi,Xj], i,j=l,2,.“,n,

where oil = V[X,] is the variance of Xi, the variance of P is given by

V[P] = ~ i: aiai,a,. i=l I=1

Since the probabilistic characteristics of P is determined by the mean E[P] = C:!, aiE[Xi] and the variance V[P], the expected utility E[u(P)] is also a function of E[P] and V[P].

Note that, for a risk-averse utility function u, E[u(P)] increases as E[P] increases while it

decreases as V[P] increases. The problem is now formulated as

(2.3)

subject to kaiE[Xt] = p, i=l

for some p. The Markowitz problem (2.3) is a quadratic programming with linear constraints,

which is not easily solved for large n. See, e.g., Konno and Kobayashi (1997) for a similar,

but tractable formulation of the portfolio selection problem, which can be solved by an

elaborated mathematical programming technique after a transformation of the problem.

When the assets X< are not normally distributed, the problem (2.1) becomes extremely

difficult to solve. Assuming that they are’ mutually independent, McEntire (1984) obtained

the following results. For a random variable X and a marginal utility function u’, define the

generalized harmonic mean (GHM) by

GH(X; u’) = ‘&(x;. (2.4)

Note that the GHM (2.4) is well defined since u’ > 0. Let S denote the choice set of assets

and assume that the assets are mutually independent. For a risk-averse utility function u, he

showed among others that none of the asset X, where X 6 S, will be included in the optimal

portfolio from the choice set 5’ U {X} if GH(Pz; u’) 2 E(X], and the expected value of any

asset included in an optimal portfolio exceeds the expected value of any asset not chosen for

the optimal portfolio. Here Pi denotes the optimal portfolio from the choice set 5’. Thus,

the structure of the optimal portfolio consists of positive amounts of the li assets, for some

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integer k, with the largest mean values. Some of these results are extended to the case that

the assets are not independent by Kijima (1997a). Th ese results are not only theoretically

interesting but also very helpful in managing portfolios with risky assets, if applicable. This

line of research would be of importance for portfolio management.

To understand the role of the GHM (2.4), we consider the two asset case (not necessarily

independent) with no short sales allowed. For two assets X and Y, the portfolio selection

problem is given by

a;i;y, +(Q); 9(a) = EMaX t (I- .P’)l (2.5)

for a risk-averse investor. Differentiation of 4(a) in (2.5) with respect to a yields

$'(a) = E[(X - Y)u'(aX + (1 - .)Y)] P.6)

and, furthermore,

#'(a) = E [(X - Y)%"(aX $ (1 - ,)Y)] (2.7)

If the utility function u is strictly concave, i.e. u”(z) < 0, as for the most of the cases in

the literature, it follows from (2.7) that $( a is strictly concave in a. Hence, if this is the )

case, the optimal fraction a* is in (0,l) if and only if $‘(a*) = 0. It follows that the optimal

fraction a* for X is zero if and only if

d’(O) = E[(X - Y),‘(Y)] < 0

or, equivalently,

WWN < W4Y)l = GH(Y; u,),

EW’)I - EM’)1 (2.8)

Therefore, when considering the situation where a* = 0, the GHM arises naturally. Ac-

cording to Wright (1987), X is said to be positive expectation dependent on Y, which we

denote by PED(X(Y), if Cov[X,y(Y)] 2 0 f or every non-decreasing function y for which

the covariance exists. Note that if X and Y are independent, we have always PED(XIY).

Suppose now PED(XIY). According to Kijima (1997a), if GH(Y; u’) 2 E[X] then a* = 0.

This is so, since if ELXl > E[XW)I

WO’)l ’ which is equivalent to Cov(X, u’(Y)] 5 0, then (2.8) holds so that n’ = 0.

Next, for a risk-averse utility function u, Arrow (1971) and Pratt (1964) considered

RA(Z; u) = -g (2.9)

as a useful risk-aversion measure. For two risk-averse utility functions u1 and ~2, we call ~2

more risk averse than ~1, denoted by ~2 &A ~1, if

Ra(z; ul) 5 Ra(z; u2) for all 2 E R. (2.10)

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Kijima and Ohnishi (1993) h s owed that, under some conditions, if us &A ur then

GH(P;;u;) 2 GH(P;;u;),

and the GHM plays a role of the risk-free threshold, where l’: denotes the optimal portfolio

for ui. We shall see another application that the GHM arises naturally in Section 5.

3 Stochastic orders and utility functions

In this section, we present stochastic orders useful in finance and provide a concise review

of their properties. It is not intended to describe the state of the art of stochastic orders.

Rather, we confine ourselves to gathering basic information needed for subsequent develop-

ments in this paper. The reader interested in details of stochastic orders should consult to,

e.g., Fomby and Seo (1989), Levy (1992), M OS er 1 and Scarsini (1993), Shaked and Shanthiku-

mar (1994), and Whitmore and Findlay (1978). I n what follows, the terms “increasing” and

“decreasing” are used to mean non-decreasing and non-increasing, respectively.

Let 3 be a class of functions on R. A stochastic ordering relation X k Y is said to be

generated from 3 if

E[f(X)] > E[f(Y)] for all f E 3. (3.1)

In this case, we denote it by X >r Y to make the dependence of 3 explicit. Note that the

stochastic ordering relation X 2~ Y depends only on the marginal distributions.

In order to describe (3.1) by an economic perspective, consider an investor with a von

Neumann-Morgenstern utility function u. Then, X 27 Y if and only if all investors whose

utility functions belong to the class 3 prefer X to Y. Recall that investors prefer more to

less. Hence the first candidate for the function class is

3~s~ = {f : f(a) is increasing in z}.

The stochastic order generated from 3~s~ is called the first order stochastic dominance

(FSD), and we denote it by X 2~s~ Y, which is just what applied probabilists call the

ordinary stochastic order. It is well known that X &so Y if and only if Fx(z) ( Fy(zc) for

all x E R, where Fx and Fy denote their respective distribution functions.

Next, investors are considered to be risk averse in the sense that their utility functions

are concave. Hence, we need to consider the function class

3Mpc = {f : f(x) is concave in z}.

The stochastic order generated from 3Mpc is called th e mean-preserving contraction (MPC),

and we denote it X >MPC Y. The MPC is usually termed as the concave order. Also, since

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in addition investors prefer more to less, the third candidate of the function class is

&sn = {f : f(z) is increasing and concave in z}.

The stochastic order generated from &sn is called the second order stochastic dominance

(SSD), and we denote it X 2 ssu Y. The SSD is usually termed as the increasing concave

order; cf. the increasing convex order defined in (1.1). It is well known that X >ssn Y if

and only if

J’ Fx(u)du 5 [ Fy(u)du, I E R. --oo -co

It should be noted that

X &PC Y M E[X] = E[Y] and X >ssn Y.

For other equivalent definitions for these stochastic ordering relations, we refer to Kijima

and Ohnishi (1996a).

Consider now the risk-aversion measure RA in (2.9). If Ra(z; u) is decreasing in Z, the

utility function u is said to display the decreasing absolzlte risk aversion (DARA). For an

investor with utility function displaying DARA, he/she becomes less risk averse as his/her

initial capital gets larger. Since a DARA utility function implies u”’ > 0, such a function

class is often used in finance. In general, the stochastic order generated from the function

class

F” = {f : (-l)“+‘f(k) > 0, k = l,.“,n}, (3.2)

where f(‘) denotes the kth order derivative off, is called the nth order stochastic dominance,

n = 1,2,.. . . For the usefulness of higher (n > 3) or d ers of stochastic dominance, we refer

to Fishburn (1980), Whitmore (1970) and references therein.

Definition 3. 1 For two random variabies X and Y, X is said to be greater than Y in the

sense of nth order stochastic dominance, if (3.1) holds for all f E ;F,,.

Each of these stochastic orders is directly related to a class of utility functions and easy

to interpret in terms of the ezpected utility principle. Namely, X 27 Y if and only if

E(u(X)] 2 E[u(Y)] for all u E .‘F. However, in the financial market, an investor is allowed

to construct a portfolio rather than simply possess an individual asset. This fact introduces

additional difficulties, and either a suitable function class or a stronger stochastic order need

be introduced. For example, according to Samuelson (1967) and Rothschild and Stiglitz

(1971) for the portfolio selection problem (2.5), if X and Y are independent and they are

equal in law, then the optimal fraction is a* = l/2 for any concave utility function u. This

is so, since the assumptions and (2.6) imply

(b’(u) = -Q(l - a), 0 5 a 2 1.

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Since #(a) is decreasing, we must have #(l/2) = 0, w ence a’ = l/2. Keeping this result in h

mind, it is natural to expect that if X 2 ~no Y and they are independent then the optimal

fraction is not less than l/2. Unfortunately, however, the result is negative and various

attempts have been made to obtain conditions sufficient for this result. For example, Hadar

and Seo (1988) confined the class of utility functions so as to give a necessary and sufficient

condition for a’ 2 l/2. We shall discuss this demand problem in the next section.

Before proceeding, we present useful stochastic orders that are related directly to no

utility function classes. They are not in common in the economic literature, but their

importance in the portfolio selection problem will become clear later. In the economic

literature, the idea has been used in, e.g., Milgrom (1981) and Jewitt (1987, 1989).

Definition 3. 2 Let X and Y be random variables on R, and suppose that they have

respective density functions fx(z) and fy(z). Then, X is said to be greater than Y in the

sense of likelihood ratio order, written by X >nnn Y, if fx(z)/fv(x) is increasing in 2 on

the support of X and Y.

Let X and Y be normally distributed with the same variance u2 and means px and py,

respectively. If PX 2 py then X >~nn Y. To see this, we have

k!& = exp

fY (xl (-~[(x--/1x)1-(2-PY)2]}’

Hence, if PX > for then, fx(z)/fv(x) is increasing in z.

Definition 3. 3 Let X and Y be random variables on R with respective distribution func-

tions F,(z) and Fy(x). Then, X is said to be greater than Y in the sense of reversed hazard

rate order, written by X >nnD Y, if F,(x)/F y z is increasing in 5 E R with the convention ( ) .

o/o = 0.

For random variable X with density function fx(s), the function TX(Z) = fx(z)/Fx(r)

is called the reversed hazard rate function of X. According to Kijima (1989), we have

X >nnn Y if and only if

m(x) 2 v(x), z E R,

whence the term “reversed hazard rate order” is justified. Recall that hx(x) = f,y(x)/Fx(x),

r;‘,(x) = 1 - Fx(x), is the hazard rate function of X, and if hx(x) < by(x) for all z then

X is said to be greater than Y in the sense of hazard rate order, written by X >hr Y. It is

well known that X >hr Y if and only if -Y >nnn -X.

So far, we have defined five stochastic orders &sn, >MPC, >ssn, &D and >nnn. Note

that these stochastic ordering relations depend only on the marginal distributions. Also, they

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are all partial ordering in the sense that they satisfy the axioms (1) XBX (reflexivity), (2)

if XBY and YBC then XBY (transitivity), and (3) if XBY and YBX then X 2 Y (anti-

symmetry), where B denotes any of the above stochastic orders and 5 stands for equality

in law. The implications of the five ordering relations are given by

x LLRD y ==+ x >RHD Y ==+ x &SD Y ===$ x &SD y.

4 The demand problem and the bivariate characteri-

zat ion

In the rest of this paper, it is assumed that the random variables under consideration have

finite expectations. Also, instead of considering the portfolio problem (2.5), we slightly

generalize it as

yy4+4 4(Q) = Eb(QX f (1 - Q)YL (4.1)

where A C R denotes some closed interval and it is assumed that l/2 E A. If no short

sales are allowed then A = [0, 11, or A = R if no constraint is imposed. However, A is not

restricted to these cases, as far as l/2 E A. In this section, we consider the demand problem

that, under what conditions, X is more demanded than Y, i.e. a’ 2 l/2 holds true.

As was shown in Section 2, our objective is to find some a for which $‘(a) = 0 where

#(CT) = E [(X - Y)u’(uX t (1 - u)Y)], a E A, (4.2)

as given in (2.6). This means that, in order to investigate the demand problem, properties

of the function defined by

f(z; y, Q) = (x - Y)“‘(UX t (1 - U)Y) (4.3)

play an essential role. Some basic properties of f(z; y, a are discussed in Kijima and Ohnishi )

(1996a).

For the demand problem, define

(4.4)

so that, from (4.2), we have

$‘(1/2) = E [(X - Y)u’ (y)] = E[Ag(X, Y)],

where

Mx, Y) = s(x, Y) - dy, 2).

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If #(l/2) 2 0, then the optimal fraction a’ cannot be less than l/2, since u” < 0. This is

why the following bivariate characterizations of stochastic orders are useful for the demand

problem.

Let G be a class of bivariate functions on R*. For two random variables X and Y, a

stochastic ordering relation X k Y is said to be generated from 6 if

E[g(X”, Y’)] 2 E[g(Y*,X*)] for all g E 9, (4.5)

where X’ and Y’ are mutually independent random variables such that X* A X and Y’ 4 Y.

In this case, we denote it by X 2~ Y. Recall that stochastic ordering relations depend only

on their marginal distributions. However, since we consider the joint distribution in (4.5),

we need to use independent random variables for the bivariate characterization. The next

two results given by Kijima and Ohnishi (1996a) are the key for the demand problem.

Theorem 4.1 Let 7 be a class of univariate functions and define

GF = {g : Ag(., y) E F for each y}.

Then, X 2~ Y if and only if X 2~~ Y.

From Theorem 4. 1 , we may want to define the following classes of bivariate functions:

GFSD = {g : Ag(s,y) is increasing in 5 for each y},

SMPC = {g : Ag(x,y) is concave in 5 for each y},

&SD = {g : Ag(z, y) is increasing and concave in 5 for each y}.

The next result is the partial converse of Theorem 4.1

Theorem 4. 2 Suppose we are given a bivariate function g(z,y) on IR’. If E[g(X,Y)] 2

E[g(Y, X)] for all X and Y such that X 2, Y and they are independent, then g E E,, where

CY stands for one of FSD, MPC or SSD.

The next result was proved by Hadar and Seo (1988), which can be easily proved by

using the bivariate characterization. Similar results also hold for MPC and SSD by the same

arguments. For details, see Kijima and Ohnishi (1996a).

Corollary 4.1 Consider the demand problem with A C IR. If X and Y are independent

random variables such that X _>FSD Y, then a* 2 l/2 for all such X and Y if and only if

2u’(Z + y) is increasing in I for any y.

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Proof. Let g(s,y) be as in (4.4) so that +‘(1/2) = E[Ag(X, I’)]. It is known that W’(I t y)

is increasing in x for any y if and only if

f(x; y, 1/2) = (x - Y)U’ (F) = 4(x, y)

is increasing in r for any y E R, where f(z;y,a) is defined by (4.3). Hence, if X &sn Y

then we have, from (4.5),

W2) = EPs(X, Y)l 2 0, which implies a* 2 l/2 since u is concave. The converse follows from Theorem 4.2.

One of the advantage to use the bivariate characterization for the demand problem is

that we can find a way of how results for independent assets are extended to the dependent

case. The next result is one of such, although this line of research is not extensive. See

Kijima and Ohnishi (1996a) for the proof.

Theorem 4.3 Consider the demand problem with A c R. Denote the joint distribution

function of (X, Y) by F(z, y). If AF(r, y) d IS ecreasing in y for y 2 z and if&(x + y) is

increasing in x for any y, then a* 2 112.

For the likelihood ratio and reversed hazard rate orders, no univariate characterization

(3.1) is known, while the following bivariate characterization is possible. The next result is

due to Shanthikumar and Yao (1991).

Theorem 4.4 Let X and Y be random variables on R:

(1) X >LRD Y if and only if (4.5) holds for

BLRD = (9: &(z,Y) L 0 for all z > Y>.

(2) X &HD Y if and only if (4.5) holds for

GRHD = {g : Ag(z, y) is increasing in x for 5 5 y}.

Note the difference between &D and r&nn. Only difference is the domain of z in which

Ag(z,y) is increasing. Hence, since &sD C &nD, we can confirm that the order >nnn is

stronger than 2~s~.

Consider the portfolio problem (2.1) with multiple assets, and assume that the constraint

A satisfies the property that (or, az, ,a,,) E A implies (a,(r), a;(s), . , a;(,)) E A, where

i(j) denote the permutation of (1,2,.. .,n). Th e ro p bl em of interest is that: Under what

conditions, we have a; 1 a; 2 _> a:. This is an extension of the demand problem to the

multiple asset case. For the extended demand problem, Theorem 4.4 plays an important

role. The next result was proved by Kijima and Ohnishi (1996a).

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Theorem 4. 5 Suppose that XI, X2,. . , X, are mutually independent

(1) If x, >LRD x2 >LRD ” LLRD xn, then a; 2 a; > > a: for every increasing utility

function u.

(2) Suppose d c [o, co)“. If x1 >RHD x2 >RHD ‘.. >RHO x,, then a; 2 a; > .” > a: foi” every increasing and concave utility function u.

According to McEntire (1984), if all the assets are independent and are ordered as

E[XJ 2 E[X,] > ... 2 E[X,,], then a: = 0 for all i such that E[Xi] < GH(P*;u’).

This result does not concern the ordering about optimal fractions. In Theorem 4. 5, the

optimal fractions a: are ordered as in the theorem, and so this result is stronger than the

McEntire’s result. We note that the research as to the comparison of demands of multiple

assets is not extensive. See Scarsini (1985, 1988) and references therein for some results of

the multiple asset case.

5 The shift effect problem

When a decision maker is faced with uncertainty concerning the economic environment, an

important question is how changes in the distribution of belief about the underlying variable

affect the optimal value of some decision variable. Typically, many authors have analyzed

the model where a decision maker wishes to maximize the expected payoff E[v(a, X))] with

respect to the decision variable a for a random variable X that represents some uncertainty,

i.e.,

F$? -Wa, XII, (5.1)

where A denotes a constraint as before. Our interest is on the effect of a shift from X

to another random variable Y, whence the problem is called the shift eflect problem. See,

e.g., Hadar and Seo (1990), Kira and Ziemba (1980), Landsberger and Meilijson (1990), and

references therein. The optimal value in (5.1) d IS enoted by ax for random variable X and

is assumed to belog to A, i.e. ax E A for any X.

As for the demand problem, the shift effect problem (5.1) can also be described in terms

of the bivariate characterization. The next result is due to Kijima and Ohnishi (1996b). In

order to explain the bivariate character of the shit effect problem, we give a concise proof.

Throughout this section, we denote

g(x,y) = u(a2,z) + u(al,y), al < a2. (5.2)

Theorem 5.1 Suppose E[Ag(X,Y)] > 0 f or all al < a2. Then, the optimal variable for

X cannot be less than that for Y, i.e. ax 2 ay.

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Proof. It is sufficient for ax 2 ay that

E[v(az, Y)] > E[v(a,, Y)] implies E[v(az, X)] > ~!$(a,, X)]. (5.3)

For, if ax < ay while (5.3) holds, then E[v(uy,X)] 2 E[v(ax,X)], since E[v(ay,Y)] 2

E[v(ax,Y)]. But, since ax is optimal for X, one has ax 2 ay by the definition of the

optimal solution, which contradicts the assumption that ax < ay. Now, in order for (5.3)

to be true, it suffices that

E(v(a,,X)] - E[v(a,, X)] L E[v(az, Y)] - EMal, Y)]

or, equivalently,

which proves the theorem.

Corollary 5. 1 Let g be a class of biwariate functions, and let g(z, y) be as in (5.2). Zf

X 20 Y and if g E G for all al < a2, then ax 2 ay.

From the above discussions, it is essential to derive the statement (5.3) for the shift effect

problem. Here is one of such examples. See Kijima and Ohnishi (1996a) for the proof and

other examples.

Theorem 5. 2 Let al < u2 and suppose that there is some x* (possibly x* = *CO) such

that ~(a~, x) 2 u(q, x) for x > x* and v(u~, z) < u(ul,.r) otherwise. Suppose X >LR~ Y. Ij

Eb(a~, Y)l L EMal, VI then Eb(a~, X)1 2 E[v(a,,X)l.

In the portfolio selection problem (2.5), the shift effect problem is given by

w(a,x) = E[u(az $ (1 - u)Z)lX = x], (5.4)

where 2 is some random variable. The next result due to Landsberger and Meilijson (1990)

can be easily proved by the bivariate characterization.

Corollary 5. 2 In the shift effect problem with A C R, let Z = r be constant. If X >LRD Y

then ax 2 uy for eve y increasing utility fundion u.

Proof. Let al < az. From (5.4), we have

w(a,x) = u(ax + (1 - u)r) = u(a(x - 7) + r).

Since u’ > 0 by our assumption, we have v(az, x) 2 U( a,, x) for x > r while v(uz, x) 5 u(~2~, x)

for I < r. Hence, the result follows from Theorems 5. 1 and 5. 2.

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Another set of the shift effect problem is as follows. Suppose that the payoff function

v(~,z) in (5.1) has the form

$a, x) = u(z(z, a)),

where u(z) is the decision maker’s utility function with u’ > 0 and U” < 0. The direct

argument ~(2, u) of the utility function represents wealth, income, or profit and is assumed

to satisfy z,,(cc, e) < 0, where z,,(z, o) denotes the second order partial derivative of Z(Z, o)

with respect to the argument a. These conditions suffice to guarantee that the first order

condition is necessary and sufficient for the optimality. This problem is often referred to

as the comparative statics for choice under risk. See, e.g., Black and Bulkley (1989), Katz

(1981), Ormiston and Schlee (1993), and references therein.

For a random variable X, define

&(a) = @44X, a))], a E A.

Let ax denote the optima1 decision variable for the shift effect problem, and assume that ax

is in the interior of A. The distribution function of X is denoted by F~(z). For simplicity,

we assume that the density function exists, which we denote by fx(~). This restriction

does not affect our results that follow and can be easily removed. Following the standard

arguments in the literature, define

Q(a) = &da) - &1(a) = l’“, +(G a))zdz, a) {fx(z) - h (x)1 dx

Then, under the assumptions given above, a~ 2 ay if and only if Q(ax) 2 0. In the

literature, various sets of conditions on the behavior of distributions of X and Y have been

derived sufficient for &(a~) > 0.

For random variable X with the density function fx(~), define

&) = ~‘(Z(~l~X))fX(~) Eb’(z(X, ax))] ’

Since u’(z) > 0, we have ix(z) > 0. Also,

whence fx(~) is a density function of some random variable X. Similarly, we can define a

random variable Y associated with Y.

Now, since ax is the optimal choice for X, we have

from which we obtain

J -1 z,(x, c&(z)dz = E(r,(& ax)] = 0.

It follows that Q(“X)

Jwz(Y, %Y))l = E [z,(&x,] - E [&,Qx,] (5.6)

Therefore, we are interested in knowing stochastic ordering relations between X and Y. For

example, if z,(z,ax) is increasing in z, then we want to have the FSD relation k 2~s~ 3.

Hereafter, to simplify the notation, we shall denote

P(5) = ‘1’(z(z, %Y)) > 0, I E R. (5.7)

Then, for random variable X, we have, from (5.5),

It follows that

which is increasing in r if so is fx(z)/fv(z), H ence, X &nn Y if and only if X >Lnn 3,

and the next result follows at once. A similar result is obtained by Ormiston and Schlee

(1993) in a different method.

Theorem 5. 3 Suppose t,,(z, a) < 0 and t,(z, ax) is increasing in 2. If X >LRD Y then

ax 2 w.

Proof X >LnD Y implies X 2~s~ 9, so we have

E [z&xl] 2 E [~a@‘, QX,]

for the increasing function z,(z,ax) in I. It follows from (5.6) that Q(a,y) > 0 and the

result follows.

For the reversed hazard rate order, a similar result has been obtained by Kijima and

Ohnishi (1996b). That is, if p(z) in (5.7) ’ d IS ecreasing in I and if X >nno Y, then .? >nHD

3. Since

b,(z) = u”(z(~, ax)Ml, ax),

we conclude that /3(z) is decreasing in I if z,(z,a) 2 0. The condition z=(+r,a) > 0,

combined with u’(z) > 0, indicates that higher values of 5 are preferred to lower for each

decision variable a, since v’(r) > 0 where V(Z) = u(z(z,a)) for each a. The next result is

derived accordingly.

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Theorem 5.4 Suppose that z,,(z, a) 5 0 and Z(Z, ax) as well as t,(z, a~) is increasing in

I. If X >RHo Y then ax > ay.

Returning to the shift effect problem with (5.4), supp ose that 2 = r is constant. This is

equivalent to assuming, in the present context, that

z(x, u) = a2 + (1 - a)r = a(r - r) $ r. (5.8)

Since, then,

ha(x, Q) = 0, 4x, a) = a, z,z(x, a) = 1,

the next result follows at once (cf. Corollary 5.2 ).

Corollary 5.3 For the shift eflect problem with A c [O,~I), let 2 = r be constant. If

X &HD Y then ax 2 ay GOT every increasing and concave function u.

A necessary and sufficient condition for ax 2 ay has been obtained by Gollier (1995) for

the case that the supports of the random variables are bounded. The bounded supports of

X and Y are denoted by Sx and Sy, respectively.

Theorem 5.5 Suppose that Sx and Sy are bounded, and that z,,(x,a) L: 0 and z,(x, a) > 0

for all x and a. Then, ax 2 ay if and only if there exists a scalar y E R such that

7 /ezm z,(t, a*)fx(t)dl > Lrn 44 a*)fY(~)d~

for all z E SX U 5’~ and every a* such that a* is the optimal fraction for &(a) for some

risk-averse utility Junction u.

Consider the case (5.8). Then, Theorem 5. 5 reveals that ax 2 ay if and only if there

exists a scalar y E R such that

7 s,Ct - r)fx(W L l:_(t - ~).fu(W,

for the given r. Of another interest would be the problem that under what conditions we

have ax 2 ay for every risk-averse utility function u and for any r. Such a problem has

been considered in Landsberger and Meilijson (1993) and subsequently by Gollier (1997).

Formally, consider the shift effect problem with

4x(a) = E[u(aX t (1 - a)r)].

Solving qSx(ax) = 0, we have

E[Xu’(axX t (1 - ax)r)]

E[u’(axX t (1 - ax)r)] = r.

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It follows that

E[Xu’(axY + (1 - u,Y)r.)] < E[Xu’(axX $ (1 - rzx)r)] E[u’(axY t (1 - ux)r)] - E[u’(axx + (1 - ax)~)]

Hence, as Landsberger and Meilijson (1993) p roved, we conclude that a~ 2 ay for every

risk-averse utility function and all r if and only if

‘WX; P) 2 GW’; P) (5.9)

for every non-negative, decreasing function b(z) = u’(axz + (1 - (IX)T), where GH(X;u’) denotes the GHM defined in (2.4). A necessary condition for (5.9) is

J’ tdFx(t) /= t@v(t)

yTx(x) > -;y(x) ’ x E R, (5.10)

or, equivalently,

E[XIX 5 x] > E[Y(Y < z], x E R. (5.11)

Unfortunately, however, the sufficient condition provided there has a flaw. Recently, Gollier

(1997) proved that a necessary and sufficient condition for (5.9) is

(5.12)

for all x < y and all p E [0, 1).

Let Pmax = suplP(t) and suppose that inf,P(t) = 0 and /&,, is finite. Since /3 is non-

negative and decreasing in our setting, p//?,,,ax can be considered as a survival function

of a random variable ‘I’, i.e. ,f?(t)/Pmax = P{T > t}. Since, for any random variable 2

independent of T, we have

EP(Z)l - = E[P{T > Z]Z}] = P{T 2 2) 13 max

and

F = E[ZP{T > Z(Z)] = E[Z; 7’ 2 21, “E.X

it follows that

The condition (5.9) is now expressed as

(5.13)

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provided that 2’ is independent of X and Y. If T = z almost surely, then we obtain (5.11).

If, onthe other hand, T has a 2-point distribution

T= {

z with probability p,

y with probability 1 - p,

then the condition (5.12) is obtained. Gollier (1997) claimed that the condition is also

sufficient.

We note that a sufficient condition for (5.9) is derived by a bivariate characterization.

Since (5.9) is equivalent to

EKX’ - y*)P(x*)P(y*)1 z 0,

if we define g(z, y) = xp(z)/3(y) so that Ag(z, y) = (2 - y)P(z)P(y), then it is equivalently

rewritten as

E[Ag(X*, Y’)] 2 0.

Since /3 > 0 and p’ < 0, we have

&=,Y) = U?=) -t (= - Y)P’(=)M(Y) 2 ‘4 = I Y.

It follows from Theorem 4. 4 (2) we conclude that if X >nno Y then (5.9) holds, as required.

According to Landsberger and Meilijson (1993), an algebra shows that (5.10) is equivalent

to the condition

J = Fx(t)dt

-03

J = Fy(t)dt

is increasing in 5. (5.14)

-co

Looking at Definitions 3.2 and 3. 3 , this might be interpreted as a higher order extension

of LRD and RHD. Let us denote X 2s Y if (5.14) holds, and recall that X >nnn Y if and

only if X >LnD Y and that if p(x) in (5.7) . 1s d ecreasing then X >nnn Y implies 2 >nnn Y.

Hence, of interest is under what conditions X 2s Y implies X 2s Y.

6 0 t her applications

In this last section, we consider some other applications of stochastic orders given in this

paper. The first example considers the price system in a complete security market, while

the second example concerns the evolution of credit rating of a firm.

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6.1 The equilibrium price system in a complete market

This subsection examines the comparative statics concerning the equilibrium price system

of a complete market. Introducing what we call Arrow-Dehreu securities and defining the

state-price, we can derive monotonicity properties of the security prices, e.g., with respect

to the shift effect of the investors’ common probabilistic belief about uncertainty.

Consider the following simple hut standard single period securities market model (see,

e.g., Duffie (1996)). Suppose that the state of economy at the end of period is classified

into one of n states R = {wr,w2,. ‘. ,w,}, and that n securities are traded in the market.

Security i is characterized by the vector zi = (zir,... ,zi,), where 5ij denotes the payoff

(or dividend) which security i pays in state w, at the end of period. We assume that the

securities market is complete, i.e., z;, i = 1,. , n, are linearly independent. Hence, without

any loss of generality, we assume that the Arrow-Debreu securities 6;, i = 1;‘. ,n, are

traded in the market, where the 6, pays 1 unit of account in state w;, and nothing elsewhere.

Suppose that there are m investors participating the market, and each investor is defined

by his/her utility function and an initial endowment. Suppose that each investor tries to

construct a portfolio maximizing his/h er expected utility from end-of-period consumption

subject to the budget constraint. Further, let A = (xi,. , A,) he the common probabilistic

belief which all investors believe, where K, > 0 he the common probabilistic assessment that

state wi occurs at the end of period. Let p = (pi,. ,p,) he the price system, where each

pi is positive and is the price of security 6, in a competitive equilibrium. Then, the market

price Q of security x = (51,. ‘, 5,), a portfolio of a,, is q = C:=r x,p,. In particular C:=, p, is the market price of a risk-free portfolio, i.e. a bond which pays 1 unit of account for sure.

Since the state price system is determined only within a multiplicative constant, without

any loss of generality, we may normalize it by C:=, e,p, = 1, where e, denotes the aggregate

consumption level in state w,.

According to Ohnishi (1999), under some regularity conditions, the price system can he

characterized by

(6.1)

k=l

where u is a utility function of the aggregated investor of the securities market with u’ > 0

and u” 5 0. From (6.1), we immediately obtain the rate of return, R say, of the risk-free

asset as

kTk{eku’(ek)}

1 + R = li=’ n (6.2)

-2x5-

Normalizing the state prices p, by (yi = p,/Cj”_, p, = (1 + R)pi, we obtain a probability

vector ~3 = (oi;.., a,), with which we have

(6.3)

for the market price of security z = (x1,. , zn). The probability vector a is called the the

risk-neutral probability.

Definition 6. 1 For two probability vectors ~9 = (ai, , ah), j = 1,2, crl is greater than

c~* in the sense of likelihood ratio order, written a1 >Lnn a*, if at/of is increasing in i on

their supports.

In the following, we assume, without any loss of generality, that states are ordered so

that er 5 . 5 e,. Since u” 5 0, we then have

u’(q) > . . . > u’(e,) > 0.

Since Ri -=K 1 -; K= ai 4 ei)

&&n{eku’(ek)l

is increasing in i, the original probability and the risk-neutral probability are ordered as

r >LRD a.

Now, let us compare the price systems under two common probability belief A’ and 79

which are ordered as 7~~ >Lnn ?r’. Let (Y’ Jr 1 + Rj, qj, j = 1,2, denote the associated risk-

neutral probability, the risk-free rate, and the market price of security z, respectively, with

the common probabilistic belief 1~‘. Since

k=l

is increasing in i, we have a1 >~un a*. Accordingly, since from (6.2)

l+Rj=~R;(lj, j=1,2, t=l

we thus have the following.

Theorem 6. 1 We have R’ > R*. Moreover, if 2; is increasing (decreasing, respectively)

in i then

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6.2 A Markov chain model of credit rating

In recent years, it becomes common to use a Markov chain model to describe the dynamics of

a firm’s credit rating as an indicator of the likelihood of default. To be more specific, consider

a time-homogeneous Markov chain {X,} on the state space N = {1,2,. , Ii, I( $ 1) with

transition matrix Q = (qv). Here state 1 represents the highest credit class, state 2 the

second highest, ., state K the lowest credit class, and state I( t 1 designates default. It

is usually assumed for simplicity that the default state h’ $ 1 is absorbing. The transition

probabilities q,, represents the actual probabilities of going from state i to state j in a unit

of time. The estimated transition probabilities can be used to predict degradation of a firm’s

credit or even its default.

In empirical studies such as Carty and Fons (1994), it is reported, among others, that

prior rating changes may carry predictive power for the direction of future rating changes.

This property is often called the rating momentum. The existence of rating momentum would

suggest that a firm upgraded (downgraded, respectively) is more likely to be subsequently

upgraded (downgraded) than downgraded (upgraded) within the next period. This empirical

finding is stated mathematically in terms of stochastic monotonicities. Namely, let {X,} be

the Markov chain defined above where Xt represents a firm’s credit rating at time t. (Note

that a higher credit rating has a smaller number in the state space.) Then, the rating

momentum is described as Xt;;-, + X{ (X,-r < Xt) implies 8, > Xlt+, (Xt < X,+1), where

Xr represents the process conditional on survival and + denotes some stochastic order. A

sufficient condition for the stochastic monotonicities of the conditional process was obtained

by Kijima (1998), which successfully explains the real data. See, e.g., Kijima (1997b) for

various notions of stochastic monotonicities of Markov chains.

The Markov chain model is used not only for describing the dynamics of a firm’s credit

rating but also for valuing risky discount bonds. The pioneering work in this direction was

done by Jarrow, Lando and Turnbull (1997) h w ere an arbitrage-free model for the term

structure of credit risk spreads was developed. The Jarrow-Lando--Turnbull model explicitly

incorporates credit rating information into the valuation methodology so as to price and

hedge options on risky debt. As the empirical research on the actual term structure of

credit risk has recognized, the credit risk spread may be increasing (decreasing, respectively)

in time for very high (low) credit quality class. This may be so, if survival increases the

probability that it has reached a low (high) credit rating. Although Kijima (1998) obtained

a sufficient condition for the credit risk spread to be monotone, an empirical study by Kijima

and Komoribayashi (1998) found that the condition does not match the real market. Hence,

theoretical conditions for the monotonicity to reflect the real market would be desired.

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